LMFDB - Embedded newform 624.4.c.e.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(337,624)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("624.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 48 $ x^4 + 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.32750i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.4.c.e.337.4

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -15.4241i q^{5} +7.96501i q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} -15.4241i q^{5} +7.96501i q^{7} +9.00000 q^{9} -12.7691i q^{11} +(7.47548 + 46.2722i) q^{13} -46.2722i q^{15} +54.0000 q^{17} -84.5794i q^{19} +23.8950i q^{21} +122.853 q^{23} -112.902 q^{25} +27.0000 q^{27} +140.853 q^{29} -116.439i q^{31} -38.3072i q^{33} +122.853 q^{35} -433.898i q^{37} +(22.4264 + 138.817i) q^{39} +205.823i q^{41} -418.853 q^{43} -138.817i q^{45} -485.861i q^{47} +279.559 q^{49} +162.000 q^{51} -674.559 q^{53} -196.951 q^{55} -253.738i q^{57} -186.226i q^{59} -671.902 q^{61} +71.6851i q^{63} +(713.706 - 115.302i) q^{65} +14.0364i q^{67} +368.559 q^{69} -346.789i q^{71} -832.900i q^{73} -338.706 q^{75} +101.706 q^{77} +335.608 q^{79} +81.0000 q^{81} +568.797i q^{83} -832.900i q^{85} +422.559 q^{87} -236.671i q^{89} +(-368.559 + 59.5423i) q^{91} -349.318i q^{93} -1304.56 q^{95} +1278.94i q^{97} -114.922i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 36 * q^9	$ 4 q + 12 q^{3} + 36 q^{9} - 72 q^{13} + 216 q^{17} - 120 q^{23} - 44 q^{25} + 108 q^{27} - 48 q^{29} - 120 q^{35} - 216 q^{39} - 1064 q^{43} - 716 q^{49} + 648 q^{51} - 864 q^{53} - 584 q^{55} - 2280 q^{61} + 1632 q^{65} - 360 q^{69} - 132 q^{75} - 816 q^{77} - 288 q^{79} + 324 q^{81} - 144 q^{87} + 360 q^{91} - 3384 q^{95}+O(q^{100}) $ 4 * q + 12 * q^3 + 36 * q^9 - 72 * q^13 + 216 * q^17 - 120 * q^23 - 44 * q^25 + 108 * q^27 - 48 * q^29 - 120 * q^35 - 216 * q^39 - 1064 * q^43 - 716 * q^49 + 648 * q^51 - 864 * q^53 - 584 * q^55 - 2280 * q^61 + 1632 * q^65 - 360 * q^69 - 132 * q^75 - 816 * q^77 - 288 * q^79 + 324 * q^81 - 144 * q^87 + 360 * q^91 - 3384 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	3.00000			0.577350
$3$	3.00000			0.577350
$4$		0			0
$5$		−	15.4241i		−	1.37957i	−0.724014	−	0.689785i	$-0.757705\pi$
$5$		−	15.4241i		−	1.37957i	0.724014	−	0.689785i	$-0.242295\pi$
$6$		0			0
$7$			7.96501i			0.430070i	0.976606	+	0.215035i	$0.0689866\pi$
$7$			7.96501i			0.430070i	−0.976606	+	0.215035i	$0.931013\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$		−	12.7691i		−	0.350002i	−0.984568	−	0.175001i	$-0.944007\pi$
$11$		−	12.7691i		−	0.350002i	0.984568	−	0.175001i	$-0.0559928\pi$
$12$		0			0
$13$	7.47548	+	46.2722i	0.159487	+	0.987200i
$13$	7.47548	+	46.2722i	0.159487	+	0.987200i
$14$		0			0
$15$		−	46.2722i		−	0.796496i
$16$		0			0
$17$	54.0000			0.770407			0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000			0.770407			0.385204	+	0.922832i	$0.374131\pi$
$18$		0			0
$19$		−	84.5794i		−	1.02126i	−0.859802	−	0.510628i	$-0.829413\pi$
$19$		−	84.5794i		−	1.02126i	0.859802	−	0.510628i	$-0.170587\pi$
$20$		0			0
$21$			23.8950i			0.248301i
$22$		0			0
$23$	122.853			1.11376			0.556882	−	0.830591i	$-0.311997\pi$
$23$	122.853			1.11376			0.556882	+	0.830591i	$0.311997\pi$
$24$		0			0
$25$	−112.902			−0.903215
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	140.853			0.901921			0.450961	−	0.892544i	$-0.351081\pi$
$29$	140.853			0.901921			0.450961	+	0.892544i	$0.351081\pi$
$30$		0			0
$31$		−	116.439i		−	0.674617i	−0.941394	−	0.337309i	$-0.890483\pi$
$31$		−	116.439i		−	0.674617i	0.941394	−	0.337309i	$-0.109517\pi$
$32$		0			0
$33$		−	38.3072i		−	0.202074i
$34$		0			0
$35$	122.853			0.593312
$36$		0			0
$37$		−	433.898i		−	1.92790i	−0.266081	−	0.963951i	$-0.585729\pi$
$37$		−	433.898i		−	1.92790i	0.266081	−	0.963951i	$-0.414271\pi$
$38$		0			0
$39$	22.4264	+	138.817i	0.0920796	+	0.569960i
$40$		0			0
$41$			205.823i			0.784003i	0.919965	+	0.392002i	$0.128217\pi$
$41$			205.823i			0.784003i	−0.919965	+	0.392002i	$0.871783\pi$
$42$		0			0
$43$	−418.853			−1.48545			−0.742726	−	0.669595i	$-0.766468\pi$
$43$	−418.853			−1.48545			−0.742726	+	0.669595i	$0.766468\pi$
$44$		0			0
$45$		−	138.817i		−	0.459857i
$46$		0			0
$47$		−	485.861i		−	1.50787i	−0.656947	−	0.753937i	$-0.728152\pi$
$47$		−	485.861i		−	1.50787i	0.656947	−	0.753937i	$-0.271848\pi$
$48$		0			0
$49$	279.559			0.815040
$50$		0			0
$51$	162.000			0.444795
$52$		0			0
$53$	−674.559			−1.74826			−0.874130	−	0.485693i	$-0.838567\pi$
$53$	−674.559			−1.74826			−0.874130	+	0.485693i	$0.838567\pi$
$54$		0			0
$55$	−196.951			−0.482852
$56$		0			0
$57$		−	253.738i		−	0.589622i
$58$		0			0
$59$		−	186.226i		−	0.410925i	−0.978665	−	0.205462i	$-0.934130\pi$
$59$		−	186.226i		−	0.410925i	0.978665	−	0.205462i	$-0.0658698\pi$
$60$		0			0
$61$	−671.902			−1.41030			−0.705149	−	0.709059i	$-0.749120\pi$
$61$	−671.902			−1.41030			−0.705149	+	0.709059i	$0.749120\pi$
$62$		0			0
$63$			71.6851i			0.143357i
$64$		0			0
$65$	713.706	−	115.302i	1.36191	−	0.220023i
$66$		0			0
$67$			14.0364i			0.0255944i	0.999918	+	0.0127972i	$0.00407358\pi$
$67$			14.0364i			0.0255944i	−0.999918	+	0.0127972i	$0.995926\pi$
$68$		0			0
$69$	368.559			0.643033
$70$		0			0
$71$		−	346.789i		−	0.579665i	−0.957077	−	0.289833i	$-0.906400\pi$
$71$		−	346.789i		−	0.579665i	0.957077	−	0.289833i	$-0.0935996\pi$
$72$		0			0
$73$		−	832.900i		−	1.33539i	−0.744435	−	0.667695i	$-0.767281\pi$
$73$		−	832.900i		−	1.33539i	0.744435	−	0.667695i	$-0.232719\pi$
$74$		0			0
$75$	−338.706			−0.521472
$76$		0			0
$77$	101.706			0.150525
$78$		0			0
$79$	335.608			0.477960			0.238980	−	0.971025i	$-0.423187\pi$
$79$	335.608			0.477960			0.238980	+	0.971025i	$0.423187\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			568.797i			0.752212i	0.926577	+	0.376106i	$0.122737\pi$
$83$			568.797i			0.752212i	−0.926577	+	0.376106i	$0.877263\pi$
$84$		0			0
$85$		−	832.900i		−	1.06283i
$86$		0			0
$87$	422.559			0.520725
$88$		0			0
$89$		−	236.671i		−	0.281877i	−0.990018	−	0.140939i	$-0.954988\pi$
$89$		−	236.671i		−	0.281877i	0.990018	−	0.140939i	$-0.0450120\pi$
$90$		0			0
$91$	−368.559	+	59.5423i	−0.424565	+	0.0685904i
$92$		0			0
$93$		−	349.318i		−	0.389491i
$94$		0			0
$95$	−1304.56			−1.40889
$96$		0			0
$97$			1278.94i			1.33873i	0.742934	+	0.669365i	$0.233434\pi$
$97$			1278.94i			1.33873i	−0.742934	+	0.669365i	$0.766566\pi$
$98$		0			0
$99$		−	114.922i		−	0.116667i
$100$		0			0
$101$	−632.264			−0.622898			−0.311449	−	0.950263i	$-0.600814\pi$
$101$	−632.264			−0.622898			−0.311449	+	0.950263i	$0.600814\pi$
$102$		0			0
$103$	1506.26			1.44094			0.720469	−	0.693487i	$-0.243927\pi$
$103$	1506.26			1.44094			0.720469	+	0.693487i	$0.243927\pi$
$104$		0			0
$105$	368.559			0.342549
$106$		0			0
$107$	1268.56			1.14613			0.573066	−	0.819509i	$-0.305754\pi$
$107$	1268.56			1.14613			0.573066	+	0.819509i	$0.305754\pi$
$108$		0			0
$109$		−	347.425i		−	0.305296i	−0.988281	−	0.152648i	$-0.951220\pi$
$109$		−	347.425i		−	0.305296i	0.988281	−	0.152648i	$-0.0487801\pi$
$110$		0			0
$111$		−	1301.69i		−	1.11307i
$112$		0			0
$113$	659.706			0.549203			0.274601	−	0.961558i	$-0.411454\pi$
$113$	659.706			0.549203			0.274601	+	0.961558i	$0.411454\pi$
$114$		0			0
$115$		−	1894.89i		−	1.53652i
$116$		0			0
$117$	67.2793	+	416.450i	0.0531622	+	0.329067i
$118$		0			0
$119$			430.111i			0.331329i
$120$		0			0
$121$	1167.95			0.877499
$122$		0			0
$123$			617.469i			0.452645i
$124$		0			0
$125$		−	186.602i		−	0.133521i
$126$		0			0
$127$	275.019			0.192157			0.0960787	−	0.995374i	$-0.469370\pi$
$127$	275.019			0.192157			0.0960787	+	0.995374i	$0.469370\pi$
$128$		0			0
$129$	−1256.56			−0.857626
$130$		0			0
$131$	1183.97			0.789648			0.394824	−	0.918757i	$-0.370805\pi$
$131$	1183.97			0.789648			0.394824	+	0.918757i	$0.370805\pi$
$132$		0			0
$133$	673.676			0.439211
$134$		0			0
$135$		−	416.450i		−	0.265499i
$136$		0			0
$137$			2557.36i			1.59482i	0.603438	+	0.797410i	$0.293797\pi$
$137$			2557.36i			1.59482i	−0.603438	+	0.797410i	$0.706203\pi$
$138$		0			0
$139$	−545.736			−0.333012			−0.166506	−	0.986040i	$-0.553249\pi$
$139$	−545.736			−0.333012			−0.166506	+	0.986040i	$0.553249\pi$
$140$		0			0
$141$		−	1457.58i		−	0.870571i
$142$		0			0
$143$	590.853	−	95.4549i	0.345522	−	0.0558205i
$144$		0			0
$145$		−	2172.52i		−	1.24426i
$146$		0			0
$147$	838.676			0.470563
$148$		0			0
$149$		−	1376.78i		−	0.756981i	−0.925605	−	0.378491i	$-0.876443\pi$
$149$		−	1376.78i		−	0.756981i	0.925605	−	0.378491i	$-0.123557\pi$
$150$		0			0
$151$		−	2733.47i		−	1.47316i	−0.676352	−	0.736579i	$-0.736440\pi$
$151$		−	2733.47i		−	1.47316i	0.676352	−	0.736579i	$-0.263560\pi$
$152$		0			0
$153$	486.000			0.256802
$154$		0			0
$155$	−1795.97			−0.930683
$156$		0			0
$157$	1029.97			0.523570			0.261785	−	0.965126i	$-0.415689\pi$
$157$	1029.97			0.523570			0.261785	+	0.965126i	$0.415689\pi$
$158$		0			0
$159$	−2023.68			−1.00936
$160$		0			0
$161$			978.524i			0.478997i
$162$		0			0
$163$			2882.91i			1.38532i	0.721264	+	0.692660i	$0.243561\pi$
$163$			2882.91i			1.38532i	−0.721264	+	0.692660i	$0.756439\pi$
$164$		0			0
$165$	−590.853			−0.278775
$166$		0			0
$167$		−	1153.90i		−	0.534679i	−0.963602	−	0.267340i	$-0.913855\pi$
$167$		−	1153.90i		−	0.534679i	0.963602	−	0.267340i	$-0.0861446\pi$
$168$		0			0
$169$	−2085.23	+	691.814i	−0.949128	+	0.314890i
$170$		0			0
$171$		−	761.215i		−	0.340418i
$172$		0			0
$173$	1688.85			0.742203			0.371101	−	0.928592i	$-0.378980\pi$
$173$	1688.85			0.742203			0.371101	+	0.928592i	$0.378980\pi$
$174$		0			0
$175$		−	899.265i		−	0.388446i
$176$		0			0
$177$		−	558.678i		−	0.237247i
$178$		0			0
$179$	−942.793			−0.393674			−0.196837	−	0.980436i	$-0.563067\pi$
$179$	−942.793			−0.393674			−0.196837	+	0.980436i	$0.563067\pi$
$180$		0			0
$181$	−482.030			−0.197950			−0.0989751	−	0.995090i	$-0.531556\pi$
$181$	−482.030			−0.197950			−0.0989751	+	0.995090i	$0.531556\pi$
$182$		0			0
$183$	−2015.71			−0.814236
$184$		0			0
$185$	−6692.47			−2.65968
$186$		0			0
$187$		−	689.530i		−	0.269644i
$188$		0			0
$189$			215.055i			0.0827670i
$190$		0			0
$191$	−4223.32			−1.59994			−0.799971	−	0.600039i	$-0.795152\pi$
$191$	−4223.32			−1.59994			−0.799971	+	0.600039i	$0.795152\pi$
$192$		0			0
$193$		−	229.092i		−	0.0854424i	−0.999087	−	0.0427212i	$-0.986397\pi$
$193$		−	229.092i		−	0.0854424i	0.999087	−	0.0427212i	$-0.0136027\pi$
$194$		0			0
$195$	2141.12	−	345.907i	0.786300	−	0.127030i
$196$		0			0
$197$			228.335i			0.0825798i	0.999147	+	0.0412899i	$0.0131467\pi$
$197$			228.335i			0.0825798i	−0.999147	+	0.0412899i	$0.986853\pi$
$198$		0			0
$199$	2939.02			1.04694			0.523471	−	0.852043i	$-0.324637\pi$
$199$	2939.02			1.04694			0.523471	+	0.852043i	$0.324637\pi$
$200$		0			0
$201$			42.1093i			0.0147769i
$202$		0			0
$203$			1121.89i			0.387889i
$204$		0			0
$205$	3174.63			1.08159
$206$		0			0
$207$	1105.68			0.371255
$208$		0			0
$209$	−1080.00			−0.357441
$210$		0			0
$211$	1607.02			0.524321			0.262161	−	0.965024i	$-0.415565\pi$
$211$	1607.02			0.524321			0.262161	+	0.965024i	$0.415565\pi$
$212$		0			0
$213$		−	1040.37i		−	0.334670i
$214$		0			0
$215$			6460.42i			2.04929i
$216$		0			0
$217$	927.441			0.290133
$218$		0			0
$219$		−	2498.70i		−	0.770988i
$220$		0			0
$221$	403.676	+	2498.70i	0.122870	+	0.760546i
$222$		0			0
$223$			130.867i			0.0392981i	0.999807	+	0.0196490i	$0.00625488\pi$
$223$			130.867i			0.0392981i	−0.999807	+	0.0196490i	$0.993745\pi$
$224$		0			0
$225$	−1016.12			−0.301072
$226$		0			0
$227$			4325.19i			1.26464i	0.774708	+	0.632319i	$0.217897\pi$
$227$			4325.19i			1.26464i	−0.774708	+	0.632319i	$0.782103\pi$
$228$		0			0
$229$		−	2621.57i		−	0.756499i	−0.925704	−	0.378250i	$-0.876526\pi$
$229$		−	2621.57i		−	0.756499i	0.925704	−	0.378250i	$-0.123474\pi$
$230$		0			0
$231$	305.117			0.0869058
$232$		0			0
$233$	4643.12			1.30550			0.652748	−	0.757575i	$-0.273616\pi$
$233$	4643.12			1.30550			0.652748	+	0.757575i	$0.273616\pi$
$234$		0			0
$235$	−7493.95			−2.08022
$236$		0			0
$237$	1006.82			0.275950
$238$		0			0
$239$			6696.69i			1.81244i	0.422809	+	0.906219i	$0.361044\pi$
$239$			6696.69i			1.81244i	−0.422809	+	0.906219i	$0.638956\pi$
$240$		0			0
$241$			2301.47i			0.615148i	0.951524	+	0.307574i	$0.0995171\pi$
$241$			2301.47i			0.615148i	−0.951524	+	0.307574i	$0.900483\pi$
$242$		0			0
$243$	243.000			0.0641500
$244$		0			0
$245$		−	4311.93i		−	1.12440i
$246$		0			0
$247$	3913.68	−	632.272i	1.00818	−	0.162876i
$248$		0			0
$249$			1706.39i			0.434290i
$250$		0			0
$251$	828.000			0.208219			0.104109	−	0.994566i	$-0.466801\pi$
$251$	828.000			0.208219			0.104109	+	0.994566i	$0.466801\pi$
$252$		0			0
$253$		−	1568.72i		−	0.389820i
$254$		0			0
$255$		−	2498.70i		−	0.613626i
$256$		0			0
$257$	−884.763			−0.214747			−0.107374	−	0.994219i	$-0.534244\pi$
$257$	−884.763			−0.214747			−0.107374	+	0.994219i	$0.534244\pi$
$258$		0			0
$259$	3456.00			0.829133
$260$		0			0
$261$	1267.68			0.300640
$262$		0			0
$263$	8343.94			1.95631			0.978155	−	0.207878i	$-0.0666557\pi$
$263$	8343.94			1.95631			0.978155	+	0.207878i	$0.0666557\pi$
$264$		0			0
$265$			10404.4i			2.41185i
$266$		0			0
$267$		−	710.013i		−	0.162742i
$268$		0			0
$269$	2762.56			0.626157			0.313078	−	0.949727i	$-0.398640\pi$
$269$	2762.56			0.626157			0.313078	+	0.949727i	$0.398640\pi$
$270$		0			0
$271$			3116.54i			0.698585i	0.937014	+	0.349293i	$0.113578\pi$
$271$			3116.54i			0.698585i	−0.937014	+	0.349293i	$0.886422\pi$
$272$		0			0
$273$	−1105.68	+	178.627i	−0.245123	+	0.0396007i
$274$		0			0
$275$			1441.65i			0.316127i
$276$		0			0
$277$	502.060			0.108902			0.0544510	−	0.998516i	$-0.482659\pi$
$277$	502.060			0.108902			0.0544510	+	0.998516i	$0.482659\pi$
$278$		0			0
$279$		−	1047.96i		−	0.224872i
$280$		0			0
$281$			6607.56i			1.40275i	0.712791	+	0.701377i	$0.247431\pi$
$281$			6607.56i			1.40275i	−0.712791	+	0.701377i	$0.752569\pi$
$282$		0			0
$283$	−4368.98			−0.917699			−0.458850	−	0.888514i	$-0.651738\pi$
$283$	−4368.98			−0.917699			−0.458850	+	0.888514i	$0.651738\pi$
$284$		0			0
$285$	−3913.68			−0.813425
$286$		0			0
$287$	−1639.38			−0.337176
$288$		0			0
$289$	−1997.00			−0.406473
$290$		0			0
$291$			3836.82i			0.772916i
$292$		0			0
$293$			5348.12i			1.06635i	0.846005	+	0.533175i	$0.179001\pi$
$293$			5348.12i			1.06635i	−0.846005	+	0.533175i	$0.820999\pi$
$294$		0			0
$295$	−2872.36			−0.566900
$296$		0			0
$297$		−	344.765i		−	0.0673579i
$298$		0			0
$299$	918.384	+	5684.67i	0.177630	+	1.09951i
$300$		0			0
$301$		−	3336.17i		−	0.638849i
$302$		0			0
$303$	−1896.79			−0.359630
$304$		0			0
$305$			10363.5i			1.94561i
$306$		0			0
$307$		−	4502.46i		−	0.837032i	−0.908210	−	0.418516i	$-0.862550\pi$
$307$		−	4502.46i		−	0.837032i	0.908210	−	0.418516i	$-0.137450\pi$
$308$		0			0
$309$	4518.79			0.831926
$310$		0			0
$311$	−7447.20			−1.35785			−0.678926	−	0.734207i	$-0.737554\pi$
$311$	−7447.20			−1.35785			−0.678926	+	0.734207i	$0.737554\pi$
$312$		0			0
$313$	−6508.93			−1.17542			−0.587710	−	0.809072i	$-0.699970\pi$
$313$	−6508.93			−1.17542			−0.587710	+	0.809072i	$0.699970\pi$
$314$		0			0
$315$	1105.68			0.197771
$316$		0			0
$317$			2465.57i			0.436846i	0.975854	+	0.218423i	$0.0700913\pi$
$317$			2465.57i			0.436846i	−0.975854	+	0.218423i	$0.929909\pi$
$318$		0			0
$319$		−	1798.56i		−	0.315674i
$320$		0			0
$321$	3805.68			0.661720
$322$		0			0
$323$		−	4567.29i		−	0.786782i
$324$		0			0
$325$	−843.996	−	5224.22i	−0.144051	−	0.891654i
$326$		0			0
$327$		−	1042.27i		−	0.176263i
$328$		0			0
$329$	3869.89			0.648491
$330$		0			0
$331$			4114.84i			0.683300i	0.939827	+	0.341650i	$0.110986\pi$
$331$			4114.84i			0.683300i	−0.939827	+	0.341650i	$0.889014\pi$
$332$		0			0
$333$		−	3905.08i		−	0.642634i
$334$		0			0
$335$	216.499			0.0353092
$336$		0			0
$337$	−4798.05			−0.775568			−0.387784	−	0.921750i	$-0.626759\pi$
$337$	−4798.05			−0.775568			−0.387784	+	0.921750i	$0.626759\pi$
$338$		0			0
$339$	1979.12			0.317082
$340$		0			0
$341$	−1486.82			−0.236117
$342$		0			0
$343$			4958.69i			0.780594i
$344$		0			0
$345$		−	5684.67i		−	0.887109i
$346$		0			0
$347$	3314.76			0.512812			0.256406	−	0.966569i	$-0.417462\pi$
$347$	3314.76			0.512812			0.256406	+	0.966569i	$0.417462\pi$
$348$		0			0
$349$			371.740i			0.0570166i	0.999594	+	0.0285083i	$0.00907570\pi$
$349$			371.740i			0.0570166i	−0.999594	+	0.0285083i	$0.990924\pi$
$350$		0			0
$351$	201.838	+	1249.35i	0.0306932	+	0.189987i
$352$		0			0
$353$			7539.10i			1.13673i	0.822776	+	0.568365i	$0.192424\pi$
$353$			7539.10i			1.13673i	−0.822776	+	0.568365i	$0.807576\pi$
$354$		0			0
$355$	−5348.89			−0.799689
$356$		0			0
$357$			1290.33i			0.191293i
$358$		0			0
$359$			12741.5i			1.87317i	0.350437	+	0.936586i	$0.386033\pi$
$359$			12741.5i			1.87317i	−0.350437	+	0.936586i	$0.613967\pi$
$360$		0			0
$361$	−294.676			−0.0429619
$362$		0			0
$363$	3503.85			0.506624
$364$		0			0
$365$	−12846.7			−1.84227
$366$		0			0
$367$	−7187.26			−1.02227			−0.511134	−	0.859501i	$-0.670774\pi$
$367$	−7187.26			−1.02227			−0.511134	+	0.859501i	$0.670774\pi$
$368$		0			0
$369$			1852.41i			0.261334i
$370$		0			0
$371$		−	5372.87i		−	0.751874i
$372$		0			0
$373$	−2087.99			−0.289845			−0.144922	−	0.989443i	$-0.546293\pi$
$373$	−2087.99			−0.289845			−0.144922	+	0.989443i	$0.546293\pi$
$374$		0			0
$375$		−	559.805i		−	0.0770886i
$376$		0			0
$377$	1052.94	+	6517.57i	0.143844	+	0.890377i
$378$		0			0
$379$		−	3982.08i		−	0.539699i	−0.962902	−	0.269850i	$-0.913026\pi$
$379$		−	3982.08i		−	0.539699i	0.962902	−	0.269850i	$-0.0869740\pi$
$380$		0			0
$381$	825.057			0.110942
$382$		0			0
$383$		−	8638.43i		−	1.15249i	−0.817278	−	0.576244i	$-0.804518\pi$
$383$		−	8638.43i		−	1.15249i	0.817278	−	0.576244i	$-0.195482\pi$
$384$		0			0
$385$		−	1568.72i		−	0.207660i
$386$		0			0
$387$	−3769.68			−0.495151
$388$		0			0
$389$	1275.74			0.166279			0.0831393	−	0.996538i	$-0.473505\pi$
$389$	1275.74			0.166279			0.0831393	+	0.996538i	$0.473505\pi$
$390$		0			0
$391$	6634.06			0.858053
$392$		0			0
$393$	3551.91			0.455904
$394$		0			0
$395$		−	5176.44i		−	0.659379i
$396$		0			0
$397$			4622.65i			0.584394i	0.956358	+	0.292197i	$0.0943862\pi$
$397$			4622.65i			0.584394i	−0.956358	+	0.292197i	$0.905614\pi$
$398$		0			0
$399$	2021.03			0.253579
$400$		0			0
$401$			138.075i			0.0171949i	0.999963	+	0.00859743i	$0.00273668\pi$
$401$			138.075i			0.0171949i	−0.999963	+	0.00859743i	$0.997263\pi$
$402$		0			0
$403$	5387.91	−	870.441i	0.665982	−	0.107592i
$404$		0			0
$405$		−	1249.35i		−	0.153286i
$406$		0			0
$407$	−5540.47			−0.674769
$408$		0			0
$409$		−	1204.64i		−	0.145637i	−0.997345	−	0.0728186i	$-0.976801\pi$
$409$		−	1204.64i		−	0.145637i	0.997345	−	0.0728186i	$-0.0231994\pi$
$410$		0			0
$411$			7672.09i			0.920770i
$412$		0			0
$413$	1483.29			0.176726
$414$		0			0
$415$	8773.16			1.03773
$416$		0			0
$417$	−1637.21			−0.192265
$418$		0			0
$419$	−5199.85			−0.606275			−0.303138	−	0.952947i	$-0.598034\pi$
$419$	−5199.85			−0.606275			−0.303138	+	0.952947i	$0.598034\pi$
$420$		0			0
$421$		−	14136.5i		−	1.63651i	−0.574854	−	0.818256i	$-0.694941\pi$
$421$		−	14136.5i		−	1.63651i	0.574854	−	0.818256i	$-0.305059\pi$
$422$		0			0
$423$		−	4372.75i		−	0.502625i
$424$		0			0
$425$	−6096.70			−0.695844
$426$		0			0
$427$		−	5351.71i		−	0.606527i
$428$		0			0
$429$	1772.56	−	286.365i	0.199487	−	0.0322280i
$430$		0			0
$431$			2279.83i			0.254793i	0.991852	+	0.127396i	$0.0406620\pi$
$431$			2279.83i			0.254793i	−0.991852	+	0.127396i	$0.959338\pi$
$432$		0			0
$433$	13298.7			1.47597			0.737984	−	0.674819i	$-0.235778\pi$
$433$	13298.7			1.47597			0.737984	+	0.674819i	$0.235778\pi$
$434$		0			0
$435$		−	6517.57i		−	0.718376i
$436$		0			0
$437$		−	10390.8i		−	1.13744i
$438$		0			0
$439$	−10452.3			−1.13635			−0.568177	−	0.822907i	$-0.692351\pi$
$439$	−10452.3			−1.13635			−0.568177	+	0.822907i	$0.692351\pi$
$440$		0			0
$441$	2516.03			0.271680
$442$		0			0
$443$	−5363.50			−0.575232			−0.287616	−	0.957746i	$-0.592863\pi$
$443$	−5363.50			−0.575232			−0.287616	+	0.957746i	$0.592863\pi$
$444$		0			0
$445$	−3650.43			−0.388870
$446$		0			0
$447$		−	4130.34i		−	0.437043i
$448$		0			0
$449$		−	9681.73i		−	1.01762i	−0.860880	−	0.508808i	$-0.830086\pi$
$449$		−	9681.73i		−	1.01762i	0.860880	−	0.508808i	$-0.169914\pi$
$450$		0			0
$451$	2628.17			0.274402
$452$		0			0
$453$		−	8200.42i		−	0.850528i
$454$		0			0
$455$	918.384	+	5684.67i	0.0946253	+	0.585718i
$456$		0			0
$457$		−	3537.94i		−	0.362139i	−0.983470	−	0.181070i	$-0.942044\pi$
$457$		−	3537.94i		−	0.362139i	0.983470	−	0.181070i	$-0.0579560\pi$
$458$		0			0
$459$	1458.00			0.148265
$460$		0			0
$461$		−	15074.9i		−	1.52302i	−0.648156	−	0.761508i	$-0.724459\pi$
$461$		−	15074.9i		−	1.52302i	0.648156	−	0.761508i	$-0.275541\pi$
$462$		0			0
$463$		−	11070.0i		−	1.11116i	−0.831463	−	0.555580i	$-0.812496\pi$
$463$		−	11070.0i		−	1.11116i	0.831463	−	0.555580i	$-0.187504\pi$
$464$		0			0
$465$	−5387.91			−0.537330
$466$		0			0
$467$	13252.8			1.31320			0.656600	−	0.754239i	$-0.271994\pi$
$467$	13252.8			1.31320			0.656600	+	0.754239i	$0.271994\pi$
$468$		0			0
$469$	−111.800			−0.0110074
$470$		0			0
$471$	3089.91			0.302284
$472$		0			0
$473$			5348.36i			0.519911i
$474$		0			0
$475$			9549.18i			0.922413i
$476$		0			0
$477$	−6071.03			−0.582753
$478$		0			0
$479$			12241.4i			1.16769i	0.811866	+	0.583843i	$0.198452\pi$
$479$			12241.4i			1.16769i	−0.811866	+	0.583843i	$0.801548\pi$
$480$		0			0
$481$	20077.4	−	3243.59i	1.90322	−	0.307474i
$482$		0			0
$483$			2935.57i			0.276549i
$484$		0			0
$485$	19726.5			1.84687
$486$		0			0
$487$			13413.3i			1.24808i	0.781392	+	0.624041i	$0.214510\pi$
$487$			13413.3i			1.24808i	−0.781392	+	0.624041i	$0.785490\pi$
$488$		0			0
$489$			8648.74i			0.799815i
$490$		0			0
$491$	−737.885			−0.0678214			−0.0339107	−	0.999425i	$-0.510796\pi$
$491$	−737.885			−0.0678214			−0.0339107	+	0.999425i	$0.510796\pi$
$492$		0			0
$493$	7606.06			0.694847
$494$		0			0
$495$	−1772.56			−0.160951
$496$		0			0
$497$	2762.17			0.249297
$498$		0			0
$499$		−	1865.65i		−	0.167370i	−0.996492	−	0.0836852i	$-0.973331\pi$
$499$		−	1865.65i		−	0.167370i	0.996492	−	0.0836852i	$-0.0266690\pi$
$500$		0			0
$501$		−	3461.70i		−	0.308697i
$502$		0			0
$503$	16632.0			1.47432			0.737161	−	0.675717i	$-0.236166\pi$
$503$	16632.0			1.47432			0.737161	+	0.675717i	$0.236166\pi$
$504$		0			0
$505$			9752.09i			0.859331i
$506$		0			0
$507$	−6255.70	+	2075.44i	−0.547979	+	0.181802i
$508$		0			0
$509$			4128.91i			0.359549i	0.983708	+	0.179775i	$0.0575368\pi$
$509$			4128.91i			0.359549i	−0.983708	+	0.179775i	$0.942463\pi$
$510$		0			0
$511$	6634.06			0.574312
$512$		0			0
$513$		−	2283.64i		−	0.196541i
$514$		0			0
$515$		−	23232.7i		−	1.98788i
$516$		0			0
$517$	−6203.99			−0.527758
$518$		0			0
$519$	5066.56			0.428511
$520$		0			0
$521$	−988.234			−0.0831005			−0.0415502	−	0.999136i	$-0.513230\pi$
$521$	−988.234			−0.0831005			−0.0415502	+	0.999136i	$0.513230\pi$
$522$		0			0
$523$	9441.62			0.789394			0.394697	−	0.918811i	$-0.370850\pi$
$523$	9441.62			0.789394			0.394697	+	0.918811i	$0.370850\pi$
$524$		0			0
$525$		−	2697.79i		−	0.224269i
$526$		0			0
$527$		−	6287.73i		−	0.519730i
$528$		0			0
$529$	2925.83			0.240472
$530$		0			0
$531$		−	1676.03i		−	0.136975i
$532$		0			0
$533$	−9523.88	+	1538.62i	−0.773968	+	0.125038i
$534$		0			0
$535$		−	19566.3i		−	1.58117i
$536$		0			0
$537$	−2828.38			−0.227288
$538$		0			0
$539$		−	3569.70i		−	0.285265i
$540$		0			0
$541$		−	14001.5i		−	1.11270i	−0.830948	−	0.556351i	$-0.812201\pi$
$541$		−	14001.5i		−	1.11270i	0.830948	−	0.556351i	$-0.187799\pi$
$542$		0			0
$543$	−1446.09			−0.114287
$544$		0			0
$545$	−5358.70			−0.421177
$546$		0			0
$547$	4244.85			0.331804			0.165902	−	0.986142i	$-0.446947\pi$
$547$	4244.85			0.331804			0.165902	+	0.986142i	$0.446947\pi$
$548$		0			0
$549$	−6047.12			−0.470100
$550$		0			0
$551$		−	11913.3i		−	0.921092i
$552$		0			0
$553$			2673.12i			0.205556i
$554$		0			0
$555$	−20077.4			−1.53556
$556$		0			0
$557$			11732.2i			0.892473i	0.894915	+	0.446236i	$0.147236\pi$
$557$			11732.2i			0.892473i	−0.894915	+	0.446236i	$0.852764\pi$
$558$		0			0
$559$	−3131.13	−	19381.2i	−0.236910	−	1.46644i
$560$		0			0
$561$		−	2068.59i		−	0.155679i
$562$		0			0
$563$	−9941.80			−0.744222			−0.372111	−	0.928188i	$-0.621366\pi$
$563$	−9941.80			−0.744222			−0.372111	+	0.928188i	$0.621366\pi$
$564$		0			0
$565$		−	10175.3i		−	0.757664i
$566$		0			0
$567$			645.166i			0.0477856i
$568$		0			0
$569$	−3690.77			−0.271925			−0.135962	−	0.990714i	$-0.543413\pi$
$569$	−3690.77			−0.271925			−0.135962	+	0.990714i	$0.543413\pi$
$570$		0			0
$571$	−5685.09			−0.416661			−0.208331	−	0.978058i	$-0.566803\pi$
$571$	−5685.09			−0.416661			−0.208331	+	0.978058i	$0.566803\pi$
$572$		0			0
$573$	−12670.0			−0.923727
$574$		0			0
$575$	−13870.3			−1.00597
$576$		0			0
$577$			7746.50i			0.558910i	0.960159	+	0.279455i	$0.0901537\pi$
$577$			7746.50i			0.558910i	−0.960159	+	0.279455i	$0.909846\pi$
$578$		0			0
$579$		−	687.275i		−	0.0493302i
$580$		0			0
$581$	−4530.47			−0.323504
$582$		0			0
$583$			8613.48i			0.611894i
$584$		0			0
$585$	6423.35	−	1037.72i	0.453971	−	0.0733410i
$586$		0			0
$587$			2766.54i			0.194527i	0.995259	+	0.0972635i	$0.0310089\pi$
$587$			2766.54i			0.194527i	−0.995259	+	0.0972635i	$0.968991\pi$
$588$		0			0
$589$	−9848.38			−0.688957
$590$		0			0
$591$			685.006i			0.0476774i
$592$		0			0
$593$			1440.79i			0.0997743i	0.998755	+	0.0498871i	$0.0158862\pi$
$593$			1440.79i			0.0997743i	−0.998755	+	0.0498871i	$0.984114\pi$
$594$		0			0
$595$	6634.06			0.457092
$596$		0			0
$597$	8817.06			0.604452
$598$		0			0
$599$	−23837.5			−1.62600			−0.813001	−	0.582263i	$-0.802168\pi$
$599$	−23837.5			−1.62600			−0.813001	+	0.582263i	$0.802168\pi$
$600$		0			0
$601$	−6694.23			−0.454348			−0.227174	−	0.973854i	$-0.572949\pi$
$601$	−6694.23			−0.454348			−0.227174	+	0.973854i	$0.572949\pi$
$602$		0			0
$603$			126.328i			0.00853145i
$604$		0			0
$605$		−	18014.6i		−	1.21057i
$606$		0			0
$607$	−3330.50			−0.222703			−0.111352	−	0.993781i	$-0.535518\pi$
$607$	−3330.50			−0.222703			−0.111352	+	0.993781i	$0.535518\pi$
$608$		0			0
$609$			3365.68i			0.223948i
$610$		0			0
$611$	22481.8	−	3632.04i	1.48857	−	0.240486i
$612$		0			0
$613$		−	13490.3i		−	0.888857i	−0.895814	−	0.444428i	$-0.853407\pi$
$613$		−	13490.3i		−	0.888857i	0.895814	−	0.444428i	$-0.146593\pi$
$614$		0			0
$615$	9523.88			0.624455
$616$		0			0
$617$			7470.76i			0.487458i	0.969843	+	0.243729i	$0.0783707\pi$
$617$			7470.76i			0.487458i	−0.969843	+	0.243729i	$0.921629\pi$
$618$		0			0
$619$			24806.9i			1.61078i	0.592746	+	0.805389i	$0.298044\pi$
$619$			24806.9i			1.61078i	−0.592746	+	0.805389i	$0.701956\pi$
$620$		0			0
$621$	3317.03			0.214344
$622$		0			0
$623$	1885.09			0.121227
$624$		0			0
$625$	−16990.9			−1.08742
$626$		0			0
$627$	−3240.00			−0.206369
$628$		0			0
$629$		−	23430.5i		−	1.48527i
$630$		0			0
$631$			314.333i			0.0198311i	0.999951	+	0.00991554i	$0.00315627\pi$
$631$			314.333i			0.0198311i	−0.999951	+	0.00991554i	$0.996844\pi$
$632$		0			0
$633$	4821.06			0.302717
$634$		0			0
$635$		−	4241.91i		−	0.265095i
$636$		0			0
$637$	2089.83	+	12935.8i	0.129988	+	0.804607i
$638$		0			0
$639$		−	3121.10i		−	0.193222i
$640$		0			0
$641$	−5550.41			−0.342009			−0.171005	−	0.985270i	$-0.554701\pi$
$641$	−5550.41			−0.342009			−0.171005	+	0.985270i	$0.554701\pi$
$642$		0			0
$643$		−	5479.48i		−	0.336065i	−0.985781	−	0.168032i	$-0.946259\pi$
$643$		−	5479.48i		−	0.336065i	0.985781	−	0.168032i	$-0.0537413\pi$
$644$		0			0
$645$			19381.2i			1.18316i
$646$		0			0
$647$	4724.83			0.287098			0.143549	−	0.989643i	$-0.454149\pi$
$647$	4724.83			0.287098			0.143549	+	0.989643i	$0.454149\pi$
$648$		0			0
$649$	−2377.93			−0.143824
$650$		0			0
$651$	2782.32			0.167508
$652$		0			0
$653$	3463.91			0.207585			0.103793	−	0.994599i	$-0.466902\pi$
$653$	3463.91			0.207585			0.103793	+	0.994599i	$0.466902\pi$
$654$		0			0
$655$		−	18261.6i		−	1.08938i
$656$		0			0
$657$		−	7496.10i		−	0.445130i
$658$		0			0
$659$	2606.35			0.154065			0.0770327	−	0.997029i	$-0.475455\pi$
$659$	2606.35			0.154065			0.0770327	+	0.997029i	$0.475455\pi$
$660$		0			0
$661$			22436.7i			1.32025i	0.751154	+	0.660127i	$0.229498\pi$
$661$			22436.7i			1.32025i	−0.751154	+	0.660127i	$0.770502\pi$
$662$		0			0
$663$	1211.03	+	7496.10i	0.0709388	+	0.439102i
$664$		0			0
$665$		−	10390.8i		−	0.605923i
$666$		0			0
$667$	17304.2			1.00453
$668$		0			0
$669$			392.600i			0.0226888i
$670$		0			0
$671$			8579.56i			0.493607i
$672$		0			0
$673$	633.970			0.0363117			0.0181558	−	0.999835i	$-0.494221\pi$
$673$	633.970			0.0363117			0.0181558	+	0.999835i	$0.494221\pi$
$674$		0			0
$675$	−3048.35			−0.173824
$676$		0			0
$677$	24457.4			1.38844			0.694221	−	0.719762i	$-0.255749\pi$
$677$	24457.4			1.38844			0.694221	+	0.719762i	$0.255749\pi$
$678$		0			0
$679$	−10186.8			−0.575747
$680$		0			0
$681$			12975.6i			0.730139i
$682$		0			0
$683$		−	12367.6i		−	0.692875i	−0.938073	−	0.346437i	$-0.887391\pi$
$683$		−	12367.6i		−	0.692875i	0.938073	−	0.346437i	$-0.112609\pi$
$684$		0			0
$685$	39445.0			2.20017
$686$		0			0
$687$		−	7864.71i		−	0.436765i
$688$		0			0
$689$	−5042.65	−	31213.3i	−0.278824	−	1.72588i
$690$		0			0
$691$		−	1050.99i		−	0.0578605i	−0.999581	−	0.0289302i	$-0.990790\pi$
$691$		−	1050.99i		−	0.0578605i	0.999581	−	0.0289302i	$-0.00921006\pi$
$692$		0			0
$693$	915.352			0.0501751
$694$		0			0
$695$			8417.46i			0.459414i
$696$		0			0
$697$			11114.4i			0.604002i
$698$		0			0
$699$	13929.4			0.753729
$700$		0			0
$701$	24294.1			1.30895			0.654476	−	0.756083i	$-0.272889\pi$
$701$	24294.1			1.30895			0.654476	+	0.756083i	$0.272889\pi$
$702$		0			0
$703$	−36698.8			−1.96888
$704$		0			0
$705$	−22481.8			−1.20101
$706$		0			0
$707$		−	5035.99i		−	0.267890i
$708$		0			0
$709$			27465.9i			1.45487i	0.686176	+	0.727436i	$0.259288\pi$
$709$			27465.9i			1.45487i	−0.686176	+	0.727436i	$0.740712\pi$
$710$		0			0
$711$	3020.47			0.159320
$712$		0			0
$713$		−	14304.9i		−	0.751365i
$714$		0			0
$715$	−1472.30	−	9113.36i	−0.0770084	−	0.476672i
$716$		0			0
$717$			20090.1i			1.04641i
$718$		0			0
$719$	36433.5			1.88976			0.944882	−	0.327411i	$-0.106176\pi$
$719$	36433.5			1.88976			0.944882	+	0.327411i	$0.106176\pi$
$720$		0			0
$721$			11997.4i			0.619704i
$722$		0			0
$723$			6904.40i			0.355156i
$724$		0			0
$725$	−15902.6			−0.814629
$726$		0			0
$727$	551.608			0.0281403			0.0140701	−	0.999901i	$-0.495521\pi$
$727$	551.608			0.0281403			0.0140701	+	0.999901i	$0.495521\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−22618.1			−1.14440
$732$		0			0
$733$		−	20317.2i		−	1.02379i	−0.859049	−	0.511893i	$-0.828945\pi$
$733$		−	20317.2i		−	1.02379i	0.859049	−	0.511893i	$-0.171055\pi$
$734$		0			0
$735$		−	12935.8i		−	0.649175i
$736$		0			0
$737$	179.232			0.00895807
$738$		0			0
$739$			29931.6i			1.48992i	0.667107	+	0.744962i	$0.267532\pi$
$739$			29931.6i			1.48992i	−0.667107	+	0.744962i	$0.732468\pi$
$740$		0			0
$741$	11741.0	−	1896.81i	0.582075	−	0.0940367i
$742$		0			0
$743$			24376.4i			1.20361i	0.798643	+	0.601805i	$0.205552\pi$
$743$			24376.4i			1.20361i	−0.798643	+	0.601805i	$0.794448\pi$
$744$		0			0
$745$	−21235.5			−1.04431
$746$		0			0
$747$			5119.17i			0.250737i
$748$		0			0
$749$			10104.1i			0.492917i
$750$		0			0
$751$	22692.2			1.10260			0.551298	−	0.834308i	$-0.314133\pi$
$751$	22692.2			1.10260			0.551298	+	0.834308i	$0.314133\pi$
$752$		0			0
$753$	2484.00			0.120215
$754$		0			0
$755$	−42161.3			−2.03232
$756$		0			0
$757$	−33063.5			−1.58747			−0.793734	−	0.608265i	$-0.791866\pi$
$757$	−33063.5			−1.58747			−0.793734	+	0.608265i	$0.791866\pi$
$758$		0			0
$759$		−	4706.15i		−	0.225062i
$760$		0			0
$761$		−	216.324i		−	0.0103045i	−0.999987	−	0.00515226i	$-0.998360\pi$
$761$		−	216.324i		−	0.0103045i	0.999987	−	0.00515226i	$-0.00164002\pi$
$762$		0			0
$763$	2767.24			0.131299
$764$		0			0
$765$		−	7496.10i		−	0.354277i
$766$		0			0
$767$	8617.09	−	1392.13i	0.405665	−	0.0655369i
$768$		0			0
$769$		−	19214.4i		−	0.901025i	−0.892770	−	0.450512i	$-0.851241\pi$
$769$		−	19214.4i		−	0.901025i	0.892770	−	0.450512i	$-0.148759\pi$
$770$		0			0
$771$	−2654.29			−0.123984
$772$		0			0
$773$		−	33175.6i		−	1.54365i	−0.635834	−	0.771826i	$-0.719344\pi$
$773$		−	33175.6i		−	1.54365i	0.635834	−	0.771826i	$-0.280656\pi$
$774$		0			0
$775$			13146.2i			0.609325i
$776$		0			0
$777$	10368.0			0.478700
$778$		0			0
$779$	17408.4			0.800667
$780$		0			0
$781$	−4428.17			−0.202884
$782$		0			0
$783$	3803.03			0.173575
$784$		0			0
$785$		−	15886.3i		−	0.722302i
$786$		0			0
$787$			14887.9i			0.674330i	0.941446	+	0.337165i	$0.109468\pi$
$787$			14887.9i			0.674330i	−0.941446	+	0.337165i	$0.890532\pi$
$788$		0			0
$789$	25031.8			1.12948
$790$		0			0
$791$			5254.56i			0.236196i
$792$		0			0
$793$	−5022.79	−	31090.4i	−0.224924	−	1.39225i
$794$		0			0
$795$			31213.3i			1.39248i
$796$		0			0
$797$	−12954.5			−0.575751			−0.287876	−	0.957668i	$-0.592949\pi$
$797$	−12954.5			−0.575751			−0.287876	+	0.957668i	$0.592949\pi$
$798$		0			0
$799$		−	26236.5i		−	1.16168i
$800$		0			0
$801$		−	2130.04i		−	0.0939591i
$802$		0			0
$803$	−10635.4			−0.467389
$804$		0			0
$805$	15092.8			0.660810
$806$		0			0
$807$	8287.68			0.361512
$808$		0			0
$809$	8275.59			0.359647			0.179823	−	0.983699i	$-0.442447\pi$
$809$	8275.59			0.359647			0.179823	+	0.983699i	$0.442447\pi$
$810$		0			0
$811$			26327.1i			1.13991i	0.821675	+	0.569956i	$0.193040\pi$
$811$			26327.1i			1.13991i	−0.821675	+	0.569956i	$0.806960\pi$
$812$		0			0
$813$			9349.63i			0.403328i
$814$		0			0
$815$	44466.3			1.91115
$816$		0			0
$817$			35426.3i			1.51703i
$818$		0			0
$819$	−3317.03	+	535.880i	−0.141522	+	0.0228635i
$820$		0			0
$821$			34439.0i			1.46398i	0.681314	+	0.731992i	$0.261409\pi$
$821$			34439.0i			1.46398i	−0.681314	+	0.731992i	$0.738591\pi$
$822$		0			0
$823$	−13870.5			−0.587479			−0.293739	−	0.955886i	$-0.594900\pi$
$823$	−13870.5			−0.587479			−0.293739	+	0.955886i	$0.594900\pi$
$824$		0			0
$825$			4324.96i			0.182516i
$826$		0			0
$827$		−	2132.30i		−	0.0896583i	−0.998995	−	0.0448292i	$-0.985726\pi$
$827$		−	2132.30i		−	0.0896583i	0.998995	−	0.0448292i	$-0.0142743\pi$
$828$		0			0
$829$	6212.39			0.260272			0.130136	−	0.991496i	$-0.458459\pi$
$829$	6212.39			0.260272			0.130136	+	0.991496i	$0.458459\pi$
$830$		0			0
$831$	1506.18			0.0628746
$832$		0			0
$833$	15096.2			0.627913
$834$		0			0
$835$	−17797.8			−0.737628
$836$		0			0
$837$		−	3143.87i		−	0.129830i
$838$		0			0
$839$			4550.52i			0.187248i	0.995608	+	0.0936242i	$0.0298452\pi$
$839$			4550.52i			0.187248i	−0.995608	+	0.0936242i	$0.970155\pi$
$840$		0			0
$841$	−4549.47			−0.186538
$842$		0			0
$843$			19822.7i			0.809880i
$844$		0			0
$845$	10670.6	+	32162.8i	0.434413	+	1.30939i
$846$		0			0
$847$			9302.74i			0.377386i
$848$		0			0
$849$	−13106.9			−0.529834
$850$		0			0
$851$		−	53305.6i		−	2.14723i
$852$		0			0
$853$			12262.8i			0.492228i	0.969241	+	0.246114i	$0.0791538\pi$
$853$			12262.8i			0.492228i	−0.969241	+	0.246114i	$0.920846\pi$
$854$		0			0
$855$	−11741.0			−0.469631
$856$		0			0
$857$	34949.1			1.39304			0.696521	−	0.717536i	$-0.254730\pi$
$857$	34949.1			1.39304			0.696521	+	0.717536i	$0.254730\pi$
$858$		0			0
$859$	21762.1			0.864394			0.432197	−	0.901779i	$-0.357739\pi$
$859$	21762.1			0.864394			0.432197	+	0.901779i	$0.357739\pi$
$860$		0			0
$861$	−4918.14			−0.194669
$862$		0			0
$863$		−	19811.5i		−	0.781450i	−0.920507	−	0.390725i	$-0.872224\pi$
$863$		−	19811.5i		−	0.781450i	0.920507	−	0.390725i	$-0.127776\pi$
$864$		0			0
$865$		−	26049.0i		−	1.02392i
$866$		0			0
$867$	−5991.00			−0.234677
$868$		0			0
$869$		−	4285.40i		−	0.167287i
$870$		0			0
$871$	−649.496	+	104.929i	−0.0252668	+	0.00408196i
$872$		0			0
$873$			11510.5i			0.446243i
$874$		0			0
$875$	1486.28			0.0574235
$876$		0			0
$877$			25716.8i			0.990186i	0.868840	+	0.495093i	$0.164866\pi$
$877$			25716.8i			0.990186i	−0.868840	+	0.495093i	$0.835134\pi$
$878$		0			0
$879$			16044.4i			0.615658i
$880$		0			0
$881$	−34709.6			−1.32735			−0.663676	−	0.748020i	$-0.731005\pi$
$881$	−34709.6			−1.32735			−0.663676	+	0.748020i	$0.731005\pi$
$882$		0			0
$883$	−3848.68			−0.146680			−0.0733400	−	0.997307i	$-0.523366\pi$
$883$	−3848.68			−0.146680			−0.0733400	+	0.997307i	$0.523366\pi$
$884$		0			0
$885$	−8617.09			−0.327300
$886$		0			0
$887$	−32804.8			−1.24180			−0.620900	−	0.783890i	$-0.713233\pi$
$887$	−32804.8			−1.24180			−0.620900	+	0.783890i	$0.713233\pi$
$888$		0			0
$889$			2190.53i			0.0826412i
$890$		0			0
$891$		−	1034.29i		−	0.0388891i
$892$		0			0
$893$	−41093.8			−1.53992
$894$		0			0
$895$			14541.7i			0.543101i
$896$		0			0
$897$	2755.15	+	17054.0i	0.102555	+	0.634802i
$898$		0			0
$899$		−	16400.8i		−	0.608452i
$900$		0			0
$901$	−36426.2			−1.34687
$902$		0			0
$903$		−	10008.5i		−	0.368840i
$904$		0			0
$905$			7434.86i			0.273086i
$906$		0			0
$907$	−22262.2			−0.814999			−0.407500	−	0.913205i	$-0.633599\pi$
$907$	−22262.2			−0.814999			−0.407500	+	0.913205i	$0.633599\pi$
$908$		0			0
$909$	−5690.38			−0.207633
$910$		0			0
$911$	13515.3			0.491528			0.245764	−	0.969330i	$-0.420961\pi$
$911$	13515.3			0.491528			0.245764	+	0.969330i	$0.420961\pi$
$912$		0			0
$913$	7263.01			0.263275
$914$		0			0
$915$			31090.4i			1.12330i
$916$		0			0
$917$			9430.33i			0.339604i
$918$		0			0
$919$	−26600.6			−0.954811			−0.477405	−	0.878683i	$-0.658423\pi$
$919$	−26600.6			−0.954811			−0.477405	+	0.878683i	$0.658423\pi$
$920$		0			0
$921$		−	13507.4i		−	0.483260i
$922$		0			0
$923$	16046.7	−	2592.41i	0.572246	−	0.0924488i
$924$		0			0
$925$			48987.9i			1.74131i
$926$		0			0
$927$	13556.4			0.480313
$928$		0			0
$929$			32887.7i			1.16148i	0.814090	+	0.580738i	$0.197236\pi$
$929$			32887.7i			1.16148i	−0.814090	+	0.580738i	$0.802764\pi$
$930$		0			0
$931$		−	23644.9i		−	0.832363i
$932$		0			0
$933$	−22341.6			−0.783956
$934$		0			0
$935$	−10635.4			−0.371993
$936$		0			0
$937$	−9261.78			−0.322913			−0.161456	−	0.986880i	$-0.551619\pi$
$937$	−9261.78			−0.322913			−0.161456	+	0.986880i	$0.551619\pi$
$938$		0			0
$939$	−19526.8			−0.678629
$940$		0			0
$941$		−	12054.9i		−	0.417619i	−0.977956	−	0.208810i	$-0.933041\pi$
$941$		−	12054.9i		−	0.417619i	0.977956	−	0.208810i	$-0.0669589\pi$
$942$		0			0
$943$			25285.9i			0.873195i
$944$		0			0
$945$	3317.03			0.114183
$946$		0			0
$947$		−	20221.4i		−	0.693885i	−0.937886	−	0.346942i	$-0.887220\pi$
$947$		−	20221.4i		−	0.693885i	0.937886	−	0.346942i	$-0.112780\pi$
$948$		0			0
$949$	38540.1	−	6226.32i	1.31830	−	0.212977i
$950$		0			0
$951$			7396.72i			0.252213i
$952$		0			0
$953$	20331.5			0.691083			0.345542	−	0.938403i	$-0.387695\pi$
$953$	20331.5			0.691083			0.345542	+	0.938403i	$0.387695\pi$
$954$		0			0
$955$			65140.8i			2.20723i
$956$		0			0
$957$		−	5395.68i		−	0.182254i
$958$		0			0
$959$	−20369.4			−0.685885
$960$		0			0
$961$	16232.9			0.544891
$962$		0			0
$963$	11417.0			0.382044
$964$		0			0
$965$	−3533.53			−0.117874
$966$		0			0
$967$		−	11082.2i		−	0.368541i	−0.982876	−	0.184270i	$-0.941008\pi$
$967$		−	11082.2i		−	0.368541i	0.982876	−	0.184270i	$-0.0589922\pi$
$968$		0			0
$969$		−	13701.9i		−	0.454249i
$970$		0			0
$971$	36694.1			1.21274			0.606369	−	0.795184i	$-0.292626\pi$
$971$	36694.1			1.21274			0.606369	+	0.795184i	$0.292626\pi$
$972$		0			0
$973$		−	4346.79i		−	0.143219i
$974$		0			0
$975$	−2531.99	−	15672.7i	−0.0831677	−	0.514797i
$976$		0			0
$977$		−	17155.3i		−	0.561767i	−0.959742	−	0.280883i	$-0.909373\pi$
$977$		−	17155.3i		−	0.561767i	0.959742	−	0.280883i	$-0.0906274\pi$
$978$		0			0
$979$	−3022.07			−0.0986575
$980$		0			0
$981$		−	3126.82i		−	0.101765i
$982$		0			0
$983$			38419.7i			1.24659i	0.781986	+	0.623296i	$0.214207\pi$
$983$			38419.7i			1.24659i	−0.781986	+	0.623296i	$0.785793\pi$
$984$		0			0
$985$	3521.86			0.113925
$986$		0			0
$987$	11609.7			0.374407
$988$		0			0
$989$	−51457.3			−1.65445
$990$		0			0
$991$	−51728.9			−1.65815			−0.829073	−	0.559140i	$-0.811131\pi$
$991$	−51728.9			−1.65815			−0.829073	+	0.559140i	$0.811131\pi$
$992$		0			0
$993$			12344.5i			0.394503i
$994$		0			0
$995$		−	45331.6i		−	1.44433i
$996$		0			0
$997$	−26846.8			−0.852805			−0.426403	−	0.904533i	$-0.640219\pi$
$997$	−26846.8			−0.852805			−0.426403	+	0.904533i	$0.640219\pi$
$998$		0			0
$999$		−	11715.2i		−	0.371025i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.c.e.337.1		4
4.3	odd	2		39.4.b.a.25.2	✓	4
12.11	even	2		117.4.b.d.64.3		4
13.12	even	2	inner	624.4.c.e.337.4		4
52.31	even	4		507.4.a.j.1.2		4
52.47	even	4		507.4.a.j.1.3		4
52.51	odd	2		39.4.b.a.25.3	yes	4
156.47	odd	4		1521.4.a.x.1.2		4
156.83	odd	4		1521.4.a.x.1.3		4
156.155	even	2		117.4.b.d.64.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.a.25.2	✓	4	4.3	odd	2
39.4.b.a.25.3	yes	4	52.51	odd	2
117.4.b.d.64.2		4	156.155	even	2
117.4.b.d.64.3		4	12.11	even	2
507.4.a.j.1.2		4	52.31	even	4
507.4.a.j.1.3		4	52.47	even	4
624.4.c.e.337.1		4	1.1	even	1	trivial
624.4.c.e.337.4		4	13.12	even	2	inner
1521.4.a.x.1.2		4	156.47	odd	4
1521.4.a.x.1.3		4	156.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -15.4241i q^{5} +7.96501i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} -15.4241i q^{5} +7.96501i q^{7} +9.00000 q^{9} -12.7691i q^{11} +(7.47548 + 46.2722i) q^{13} -46.2722i q^{15} +54.0000 q^{17} -84.5794i q^{19} +23.8950i q^{21} +122.853 q^{23} -112.902 q^{25} +27.0000 q^{27} +140.853 q^{29} -116.439i q^{31} -38.3072i q^{33} +122.853 q^{35} -433.898i q^{37} +(22.4264 + 138.817i) q^{39} +205.823i q^{41} -418.853 q^{43} -138.817i q^{45} -485.861i q^{47} +279.559 q^{49} +162.000 q^{51} -674.559 q^{53} -196.951 q^{55} -253.738i q^{57} -186.226i q^{59} -671.902 q^{61} +71.6851i q^{63} +(713.706 - 115.302i) q^{65} +14.0364i q^{67} +368.559 q^{69} -346.789i q^{71} -832.900i q^{73} -338.706 q^{75} +101.706 q^{77} +335.608 q^{79} +81.0000 q^{81} +568.797i q^{83} -832.900i q^{85} +422.559 q^{87} -236.671i q^{89} +(-368.559 + 59.5423i) q^{91} -349.318i q^{93} -1304.56 q^{95} +1278.94i q^{97} -114.922i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 36 * q^9	\( 4 q + 12 q^{3} + 36 q^{9} - 72 q^{13} + 216 q^{17} - 120 q^{23} - 44 q^{25} + 108 q^{27} - 48 q^{29} - 120 q^{35} - 216 q^{39} - 1064 q^{43} - 716 q^{49} + 648 q^{51} - 864 q^{53} - 584 q^{55} - 2280 q^{61} + 1632 q^{65} - 360 q^{69} - 132 q^{75} - 816 q^{77} - 288 q^{79} + 324 q^{81} - 144 q^{87} + 360 q^{91} - 3384 q^{95}+O(q^{100}) \) 4 * q + 12 * q^3 + 36 * q^9 - 72 * q^13 + 216 * q^17 - 120 * q^23 - 44 * q^25 + 108 * q^27 - 48 * q^29 - 120 * q^35 - 216 * q^39 - 1064 * q^43 - 716 * q^49 + 648 * q^51 - 864 * q^53 - 584 * q^55 - 2280 * q^61 + 1632 * q^65 - 360 * q^69 - 132 * q^75 - 816 * q^77 - 288 * q^79 + 324 * q^81 - 144 * q^87 + 360 * q^91 - 3384 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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